This lecture, moved to 3 rue d'Ulm, has not been recorded.
Abstract
We continue our study of large random regular graphs by describing their diameters. We also prove B. Bollobás' theorem : the number of closed geodesics of given length on such a graph asymptotically follows a Poisson distribution.
We then briefly introduce another model of random graphs, constructed by random coverings, before moving on to hyperbolic surfaces. In this lecture, we consider the Brooks-Makover model obtained by randomly gluing together ideal hyperbolic triangles.
References
B. Bollobás, Random graphs, Cambridge University Press. Proof of Corollary 2.19. Chapter 10 : diameter of large random regular graphs.
C. Bordenave, B. Collins, Eigenvalues of random lifts and polynomials of random permutation matrices, Annals of Math. 2019. Random covering model : paragraph 1.5.
R. Brooks, E. Makover, Random Construction of Riemann Surfaces. J. Diff. Geom. 2004, statement of main results.
M. Magee, letter to B. Petri, https://www.mmagee.net/diameter.pdf : link between spectral hole and diameter of a hyperbolic surface.
B. Petri, Random regular graphs and the systole of a random surface, J. Topology 2017, statement of theorem B.
